= Solution
The <orthogonal complement> is
$$
\boxed{Y^\perp=\{w\in X:\langle w,y\rangle=0\text{ for every }y\in Y\}.}
$$
It is a <linear subspace> and is closed, since it is an intersection of <kernels> of the continuous <linear functionals> $w\mapsto\langle w,y\rangle$. Continuity follows from the <Cauchy-Schwarz inequality>.
The minimizing point and orthogonality proved below write every $x$ as $x=y_0+(x-y_0)\in Y+Y^\perp$. If $w\in Y\cap Y^\perp$, then $\|w\|^2=\langle w,w\rangle=0$, so the intersection is zero. Consequently
$$
\boxed{X=Y\oplus Y^\perp.}
$$
In particular the minimizing point is unique: an alternative orthogonal decomposition would have a difference in this zero intersection.
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